ЦТ Mathematics common mistakes
ЦТ Mathematics · Understand the traps and how to avoid them.
Simplifies an even root of a square as √(x²)=x and then cancels or evaluates expressions without checking the sign of x.
The learner memorizes the root rule as an unconditional identity and forgets that √(x²)=|x| for real x, so negative values silently change the expression.
What to do instead
Before simplifying any even root, rewrite it with an absolute value, then split into cases for x≥0 and x<0. Test the simplified result with one negative and one positive numerical value.
Solves a rational or radical equation by clearing denominators or squaring both sides and accepts every algebraic result without checking it in the original equation.
Squaring and multiplying by expressions that can be zero are not reversible operations, so extraneous roots are created; learners focus on the transformed equation and forget the original domain.
What to do instead
Write all restrictions first: denominators nonzero, radicands nonnegative where required. After solving, substitute each candidate into the original equation and reject any that violate a restriction or fail equality.
Multiplies or divides an inequality by a variable expression without considering its sign, or writes a solution interval with the wrong bracket for a strict inequality.
The learner treats inequalities like equations and does not track that multiplying by a negative number reverses the inequality sign; endpoint inclusion is often copied mechanically.
What to do instead
Move all terms to one side, factor, and use a sign chart with test points. Never divide by an expression whose sign is unknown. Mark excluded points as open and check whether each endpoint is allowed by the original inequality.
Reads y=f(x+2) as a shift two units to the right, or states the domain and range of a transformed function by shifting both in the same direction without solving for the new input restrictions.
Horizontal transformations are remembered visually rather than from the replacement of x, and domain restrictions are treated as labels attached to the graph instead of inequalities in the new variable.
What to do instead
Evaluate key points: if x=0 in the parent gives f(0), then x=-2 in f(x+2) gives the same output, so the graph shifts left. For domain, solve the inequality that keeps the original input inside the parent domain; for range, track the output transformation.
Divides a trigonometric equation by sin x or cos x and loses whole families of solutions, or solves sin x=a using only x=arcsin a+2πk and omits x=π−arcsin a+2πk.
The learner treats inverse trigonometric functions as if they returned all solutions, and division by a trigonometric factor hides the case where that factor is zero.
What to do instead
Never divide by a trigonometric expression; factor and set each factor to zero. Use the unit circle or general solution formulas, including both symmetric families for sine and cosine, and check the period.
Uses an arithmetic-series sum formula with the wrong number of terms, confuses a_n with S_n, or applies the infinite geometric sum formula without checking that |q|<1.
Formulas are recalled as strings without their definitions and convergence conditions, and the count of terms is guessed from the last index rather than computed.
What to do instead
Write the definitions a_n=a_1+(n−1)d and S_n=n(a_1+a_n)/2, then compute n explicitly from the first and last indices. For geometric series, state the common ratio and verify |q|<1 before using S=a_1/(1−q).
Applies a linear scale factor to areas or volumes of similar figures, for example multiplying an area by k when the side lengths are multiplied by k.
The learner remembers that similar figures have proportional sides but does not distinguish one-dimensional, two-dimensional, and three-dimensional measures.
What to do instead
Label the similarity ratio as k. Then lengths scale by k, areas by k², and volumes by k³. Before computing, identify whether the requested quantity is a length, area, or volume and apply the matching power.
Uses the slant height of a cone or pyramid as the perpendicular height in a volume formula, or omits the factor 1/3 when calculating the volume of a pyramid or cone.
Two-dimensional drawings make slant height and perpendicular height look interchangeable, and surface-area formulas are mixed with volume formulas.
What to do instead
Draw the right triangle containing the perpendicular height, slant height, and radius or apothem. Use the Pythagorean relation to find the true height. Write the volume formula with the 1/3 factor and check that the result has cubic units.
Counts ordered arrangements when the problem asks for groups, or adds probabilities of overlapping events without subtracting the intersection.
The learner does not explicitly model the sample space, so the distinction between order mattering and not mattering, and between mutually exclusive and overlapping events, is lost.
What to do instead
Define the experiment and label whether order matters before choosing a counting method. For probability, draw a Venn diagram or write P(A∪B)=P(A)+P(B)−P(A∩B) and identify whether the intersection is empty.
Sets up a percentage, mixture, motion, or work equation using the wrong base or mismatched units, such as adding times instead of rates when two workers act together.
The learner translates words directly into symbols without tabulating the quantities, and does not check whether the units of each term are compatible.
What to do instead
Make a table with columns for rate, time, amount, or concentration, and write units beside every entry. For work, add rates, not times. For percentages, state the base explicitly. Check dimensional consistency and substitute the answer back into the word problem.